On Saab's Characterizations of Weak Radon–Nikodym Sets

Abstract

Throughout this paper X and Y denote real Banach spaces with topological duals X* and F* respectively. The closed unit ball in X is denoted by Bx. Recently, in parallel with that of dual Banach spaces with the Radon-Nikodym property (RNP), the study of such spaces with the weak Radon-Nikodym property (WRNP) as well as Banach spaces not containing a copy of ^ has been made by many authors, especially, Pelczynski [14], Rosenthal [18], [19], Odell and Rosenthal [13], Haydon [7], Musial [11], Janicka [9], Riddle and Uhr [17], and Saab and Saab [23]. Corresponding to those of dual Banach spaces with the RNP, a number of characterizations of such spaces with the WRNP have been obtained, heavily relying on Rosenthal's signal theorem (Theorem 1 in [18] or Theorem 2.2 in [19]) asserting that the space X contains no copy of /i if and only if every bounded subset of X is weakly pre-compact (For this terminology, see § 3). They are collected below.